Looking at Bessel Functions & Applications

Plot of Bessel function of the first kind, J_α(x), for integer orders α = 0, 1, 2.

Among the most important special functions is the Bessel function. In the field of solar physics, for example, it's of inestimable importance in the analysis of solar magnetic fields and their evolution. One very key equation is the Helmholtz, viz.

1/ r  [¶/ ¶r  ( r  ¶/ ¶r)] B  +  (a)² B = 0

where r is the radial coordinate, B  the magnetic field intensity, and a a quantity called the "force free parameter". Then the axially symmetric (i.e.- in cylindrical coordinates r, z, q) Bessel function solutions are

B _z (r)    =   B_o J_o(a r)  

B _q (r)  =  B_o J₁(ar)

where the axial (top) and azimuthal magnetic field components are given, respectively, and  J_o(a r) is a Bessel function of the first kind, order zero and J₁(ar) is a Bessel function of the first kind, order unity. (See graphs at the top for Bessel functions of the orders 0, 1 and 2).

The Bessel functions are mathematically defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)…(m + j) +  …]

which we terminate with second order terms.

For m = 0 and m = 1 forms one gets:

J_o(x) = 1 - x²/ 2² (1!)² + x⁴/ 2⁴ (2!)² - x⁶/ 2⁶ (3!)² + ......

and:

J₁(x) = x/ 2 - x³/ 2³ ·1! 2! + x⁵/ 2⁵ ·2!3! - x⁷/ 2⁷ ·3!4! - .....

The equations in B _z (r),  B _q (r),  with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube. For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r)  =  (L B _q(r))/  (r  B _z (r))

Where L denotes the length of the sunspot-flux tube dipole and r, the radius. If the twist exceeds 2p then the magnetic configuration may be approaching instability and a solar flare.

Problems for the Math Maven:

1) Compute the Bessel functions for J_o(x)   and  J₁(x)  with x = 1 and then compare with the values obtained from the graph shown at top.

2) Find the twist in a solar loop (take it to be a magnetic tube) if: B _q(r) = 0. 1T and B _z (r) = 0.2T. Take the radius of the tube to be r = 10 ⁴  km and the length L = 10 ⁸  m.   Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) > 2p)

3) Compute the intensity for the azimuthal magnetic field component (i.e. B _q (r) ) of a large sunspot, if its equilibrium magnetic field B_o = 0.01 T and the value of  J₁(ar) conforms to a = 0.4 and r = 40.